Mathematische Zeitschrift

, Volume 271, Issue 3–4, pp 1211–1239

On stochastic completeness of jump processes

  • Alexander Grigor’yan
  • Xueping Huang
  • Jun Masamune
Article

Abstract

We prove the following sufficient condition for stochastic completeness of symmetric jump processes on metric measure spaces: if the volume of the metric balls grows at most exponentially with radius and if the distance function is adapted in a certain sense to the jump kernel then the process is stochastically complete. We use this theorem to prove the following criterion for stochastic completeness of a continuous time random walk on a graph with a counting measure: if the volume growth with respect to the graph distance is at most cubic then the random walk is stochastically complete, where the cubic volume growth is sharp.

Keywords

Jump processes Random walks Stochastic completeness Non-local Dirichlet forms Physical Laplacian 

Mathematics Subject Classification (2000)

Primary 60J75 Secondary 60J25 60J27 05C81 

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Copyright information

© Springer-Verlag 2011

Authors and Affiliations

  • Alexander Grigor’yan
    • 1
  • Xueping Huang
    • 1
  • Jun Masamune
    • 2
  1. 1.Department of MathematicsUniversity of BielefeldBielefeldGermany
  2. 2.Department of Mathematics and StatisticsPennsylvania State UniversityAltoonaUSA

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